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Unformatted text preview: Riemann Surfaces See Arfken & Weber section 6.7 (mapping) or [1][sections 106-108] for more information. 1 Definition A Riemann surface is a generalization of the complex plane to a surface of more than one sheet such that a multiple-valued function on the original complex plane has only one value at each point of the surface. Briefly, the notion of a Riemann surface is important whenever considering functions with branch cuts. Perhaps the easiest example is the Riemann surface for log z . Write z = r exp( iθ ), then log z = (log r ) + iθ . However, the θ in that expression is not uniquely defined – it is only defined up to a multiple of 2 π . As you walk around the origin of the complex plane, the function log z comes back to itself up to additions of 2 πi . We can construct a cover of the complex plane on which log z is single-valued by taking a helix over the complex plane – going 360 ◦ about the origin takes from you from sheet of the cover to another sheet. Put another way, every time you go through the branch cut, you go to a different sheet. We can construct such a cover as follows. (Our discussion here is verbatim from [1][section 106].) Consider the complex z plane, with the origin deleted, as a sheet R which is cut along the positive half of the real axis. On that sheet, let θ range from 0 to 2 π . Let a second sheet R 1 be cut in the same way and placed in front of the sheet R . The lower edge of the slit in R is joined to the upper edge of the slit in R 1 . On R 1 ,...
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